Notes

As an analog of the uncertainty principle, one of the implications of the basic formalism of quantum theory is that an ordinary quantum field can in a sense never maintain precisely zero value, but must always show certain fluctuations—even in what one considers the vacuum. And in terms of Feynman diagrams the way this happens is by virtual particle-antiparticle pairs of all types and all energy-momenta continually forming and annihilating at all points in the vacuum. Insofar as such vacuum fluctuations are always exactly the same, however, they presumably cannot be detected. (In the formalism of quantum field theory, they are usually removed by so-called normal ordering. But without this every mode of any quantum system will show a zero-point energy 1/2 ℏ ω—positive in sign for bosons and negative for fermions, cancelling for perfect supersymmetry. Quite what gravitational effects such zero-point energy might have has never been clear.) If one somehow changes the space in which a vacuum exists, there can be directly observable effects of vacuum fluctuations. An example is the 1948 Casimir effect—in which the absence of low-energy (long wavelength) virtual particle pairs in the space between two metal plates (but not in the infinite space outside) leads to a small but measurable force of attraction between them. The different detailed patterns of modes of different fields in different spaces can lead to very different effective vacuum energies—often negative. And at least with the idealization of impermeable classical conducting boundaries one predicts (based on work of mine from 1981) the peculiar effect that closed cycles can be set up that systematically extract energy from vacuum fluctuations in a photon field.

If one has moving boundaries it turns out that vacuum fluctuations can in effect be viewed as producing real particles. And as known since the 1960s, the same is true for expanding universes. What happens in essence is that the modes of fields in different background spacetime structures differ to the point where zero-point excitations seem like actual particle excitations to a detector or observer calibrated to fields in ordinary fixed flat infinite spacetime. And in fact just uniform acceleration turns out to make detectors register real particles in a vacuum—in this case with a thermal spectrum at a temperature proportional to the acceleration. (Uniform rotation also leads to real particles, but apparently with a different spectrum.) As expected from the equivalence principle, a uniform gravitational field should produce the same effect. (Uniform electric fields lead in a formally similar way to production of charged particles.) And as pointed out by Stephen Hawking in 1974, black holes should also generate thermal radiation (at a temperature (ℏ c3)/(8 π G k M)). A common interpretation is that the radiated particles are somehow ones left behind when the other particle in a virtual pair goes inside the event horizon. (A similar explanation can be given for uniform acceleration—for which there is also an event horizon.) There has been much discussion of the idea that Hawking radiation somehow shows pure quantum states spontaneously turning into mixed ones, more or less as in quantum measurements. But presumably this is just a reflection of the idealization involved in looking at quantum fields in a fixed background classical spacetime. And indeed work in string theory in the mid-1990s may suggest ways in which quantum gravity configurations of black hole surfaces could maintain the information needed for the whole system to act as a pure state.